Einstein metrics from Derdziński duality

Abstract：

(joint work with Rosa Sena-Dias) A theorem of Derdziński from the 1980's establishes that certain Einstein metrics are conformal to Bach-flat extremal Kahler metrics. Using this result Rosa Sena-Dias and I classified conformally Kähler, U(2)-invariant, Einstein metrics on the total space of O(−m). This yields infinitely many 1-parameter families of metrics exhibiting several different behaviors including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface with a cone singularity along the ''divisor at infinity''. As an application of these results, we find many interesting phenomena. For instance, we exhibit the Taub-bolt Ricci-flat ALF metric as a limit of cone angle Einstein metrics on the blow up of CP2 at a point (in the limit when the cone angle converges to zero). We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric and cannot be obtained by applying Derdziński's theorem.